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Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations. (English) Zbl 1142.11049

The authors study a “mixed” simultaneous inhomogeneous approximation problem in the sense that the key space is \(\Omega=\mathbb{R}\times\mathbb{C}\mathbb{Q}_p\) where \(\mathbb{Q}_p\) is \(p\)-adic space for some prime \(p\). Each of these spaces has a natural measure associated to it and the authors then study a classical problem in metric number theory with respect to the measure \(\mu\) on \(\Omega \) which is defined to be the measure obtained by taking the product of each of the natural measures of the component sets \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{Q}_p\). The authors show that a system of inhomogeneous inequalities, each of the form \(| P_n(x_i) + d_i| < H^{\lambda_i}\psi^{\nu_i} \), holds simultaneously for \(i=1,2,3\) only for a finite number of polynomials \(P_n\in\mathbb{Z}[X]\) of degree \(n\) for any (fixed) point \(d\in\Omega\) and almost all (with respect to the measure \(\mu\)) \((x_1,x_2,x_3)\in\Omega\). Here \(H\) is the height of the polynomial \(P_n\) and there are certain conditions imposed on the values \(\lambda_i\) and \(\nu_i\). The proof relies on a development of Sprindzhuk’s notion of essential and inessential domains. Essentially the authors consider a number of cases where the value of \(P_n^\prime\) is large or small and use various technical lemmas to arrive at the main theorem of the paper.

MSC:

11J61 Approximation in non-Archimedean valuations
11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
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References:

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